Refractive index

In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium.

The refractive index determines how much the path of light is bent, or refracted, when entering a material. This is described by Snell's law of refraction, $n 1 sin θ 1 = n 2 sin θ 2$ , where $θ 1$ and $θ 2$ are the angle of incidence and angle of refraction, respectively, of a ray crossing the interface between two media with refractive indices $n 1$ and $n 2$ . The refractive indices also determine the amount of light that is reflected when reaching the interface, as well as the critical angle for total internal reflection, their intensity (Fresnel equations) and Brewster's angle.^[1]

The refractive index, $n$ , can be seen as the factor by which the speed and the wavelength of the radiation are reduced with respect to their vacuum values: the speed of light in a medium is $v = c/ n$ , and similarly the wavelength in that medium is $λ = λ 0 / n$ , where $λ 0$ is the wavelength of that light in vacuum. This implies that vacuum has a refractive index of 1, and assumes that the frequency ( $f = v / λ$ ) of the wave is not affected by the refractive index.

The refractive index may vary with wavelength. This causes white light to split into constituent colors when refracted. This is called dispersion. This effect can be observed in prisms and rainbows, and as chromatic aberration in lenses. Light propagation in absorbing materials can be described using a complex-valued refractive index.^[2] The imaginary part then handles the attenuation, while the real part accounts for refraction. For most materials the refractive index changes with wavelength by several percent across the visible spectrum. Consequently, refractive indices for materials reported using a single value for $n$ must specify the wavelength used in the measurement.

The concept of refractive index applies across the full electromagnetic spectrum, from X-rays to radio waves. It can also be applied to wave phenomena such as sound. In this case, the speed of sound is used instead of that of light, and a reference medium other than vacuum must be chosen.^[3]

For lenses (such as eye glasses), a lens made from a high refractive index material will be thinner, and hence lighter, than a conventional lens with a lower refractive index. Such lenses are generally more expensive to manufacture than conventional ones.

Definition

The relative refractive index of an optical medium 2 with respect to another reference medium 1 ( $n 21$ ) is given by the ratio of speed of light in medium 1 to that in medium 2. This can be expressed as follows:

n_{21}={\frac {v_{1}}{v_{2}}}.

If the reference medium 1 is vacuum, then the refractive index of medium 2 is considered with respect to vacuum. It is simply represented as

n 2

and is called the absolute refractive index of medium 2.

The absolute refractive index n of an optical medium is defined as the ratio of the speed of light in vacuum, $c = 299792458 m/s$ , and the phase velocity $v$ of light in the medium,

n={\frac {\mathrm {c} }{v}}.

Since

c

is constant,

n

is inversely proportional to

v

:

n\propto {\frac {1}{v}}.

The phase velocity is the speed at which the crests or the phase of the wave moves, which may be different from the group velocity, the speed at which the pulse of light or the envelope of the wave moves.^[1] Historically air at a standardized pressure and temperature has been common as a reference medium.

History

Thomas Young was presumably the person who first used, and invented, the name "index of refraction", in 1807.^[4] At the same time he changed this value of refractive power into a single number, instead of the traditional ratio of two numbers. The ratio had the disadvantage of different appearances. Newton, who called it the "proportion of the sines of incidence and refraction", wrote it as a ratio of two numbers, like "529 to 396" (or "nearly 4 to 3"; for water).^[5] Hauksbee, who called it the "ratio of refraction", wrote it as a ratio with a fixed numerator, like "10000 to 7451.9" (for urine).^[6] Hutton wrote it as a ratio with a fixed denominator, like 1.3358 to 1 (water).^[7]

Young did not use a symbol for the index of refraction, in 1807. In the later years, others started using different symbols: $n$ , $m$ , and $µ$ .^[8]^[9]^[10] The symbol $n$ gradually prevailed.

Typical values

Gemstone diamonds — Diamonds have a very high refractive index of 2.417.

Refractive index also varies with wavelength of the light as given by Cauchy's equation:

The most general form of Cauchy's equation is

n(\lambda )=A+{\frac {B}{\lambda ^{2}}}+{\frac {C}{\lambda ^{4}}}+\cdots ,

where

n

is the refractive index,

λ

is the wavelength, and

A

,

B

,

C

, etc., are coefficients that can be determined for a material by fitting the equation to measured refractive indices at known wavelengths. The coefficients are usually quoted for

λ

as the vacuum wavelength in micrometres.

Usually, it is sufficient to use a two-term form of the equation:

n(\lambda )=A+{\frac {B}{\lambda ^{2}}},

where the coefficients

A

and

B

are determined specifically for this form of the equation.

Selected refractive indices at $λ$ = 589 nm. For references, see the extended List of refractive indices.
Material	$n$
Vacuum	1
Gases at 0 °C and 1 atm
Air	1.000293
Helium	1.000036
Hydrogen	1.000132
Carbon dioxide	1.00045
Liquids at 20 °C
Water	1.333
Ethanol	1.36
Olive oil	1.47
Solids
Ice	1.31
Fused silica (quartz)	1.46^[11]
PMMA (acrylic, plexiglas, lucite, perspex)	1.49
Window glass	1.52^[12]
Polycarbonate (Lexan™)	1.58^[13]
Flint glass (typical)	1.69
Sapphire	1.77^[14]
Cubic zirconia	2.15
Diamond	2.417
Moissanite	2.65

For visible light most transparent media have refractive indices between 1 and 2. A few examples are given in the adjacent table. These values are measured at the yellow doublet D-line of sodium, with a wavelength of 589 nanometers, as is conventionally done.^[15] Gases at atmospheric pressure have refractive indices close to 1 because of their low density. Almost all solids and liquids have refractive indices above 1.3, with aerogel as the clear exception. Aerogel is a very low density solid that can be produced with refractive index in the range from 1.002 to 1.265.^[16] Moissanite lies at the other end of the range with a refractive index as high as 2.65. Most plastics have refractive indices in the range from 1.3 to 1.7, but some high-refractive-index polymers can have values as high as 1.76.^[17]

For infrared light refractive indices can be considerably higher. Germanium is transparent in the wavelength region from 2 to 14 μm and has a refractive index of about 4.^[18] A type of new materials termed "topological insulators", was recently found which have high refractive index of up to 6 in the near to mid infrared frequency range. Moreover, topological insulators are transparent when they have nanoscale thickness. These properties are potentially important for applications in infrared optics.^[19]

Refractive index below unity

According to the theory of relativity, no information can travel faster than the speed of light in vacuum, but this does not mean that the refractive index cannot be less than 1. The refractive index measures the phase velocity of light, which does not carry information.^[20]^[a] The phase velocity is the speed at which the crests of the wave move and can be faster than the speed of light in vacuum, and thereby give a refractive index below 1. This can occur close to resonance frequencies, for absorbing media, in plasmas, and for X-rays. In the X-ray regime the refractive indices are lower than but very close to 1 (exceptions close to some resonance frequencies).^[21] As an example, water has a refractive index of 0.99999974 = 1 − 2.6×10⁻⁷ for X-ray radiation at a photon energy of 30 keV (0.04 nm wavelength).^[21]

An example of a plasma with an index of refraction less than unity is Earth's ionosphere. Since the refractive index of the ionosphere (a plasma), is less than unity, electromagnetic waves propagating through the plasma are bent "away from the normal" (see Geometric optics) allowing the radio wave to be refracted back toward earth, thus enabling long-distance radio communications. See also Radio Propagation and Skywave.^[22]

Negative refractive index

Recent research has also demonstrated the "existence" of materials with a negative refractive index, which can occur if permittivity and permeability have simultaneous negative values.^[23] This can be achieved with periodically constructed metamaterials. The resulting negative refraction (i.e., a reversal of Snell's law) offers the possibility of the superlens and other new phenomena to be actively developed by means of metamaterials.^[24]^[25]

Microscopic explanation

At the atomic scale, an electromagnetic wave's phase velocity is slowed in a material because the electric field creates a disturbance in the charges of each atom (primarily the electrons) proportional to the electric susceptibility of the medium. (Similarly, the magnetic field creates a disturbance proportional to the magnetic susceptibility.) As the electromagnetic fields oscillate in the wave, the charges in the material will be "shaken" back and forth at the same frequency.^[1]^: 67 The charges thus radiate their own electromagnetic wave that is at the same frequency, but usually with a phase delay, as the charges may move out of phase with the force driving them (see sinusoidally driven harmonic oscillator). The light wave traveling in the medium is the macroscopic superposition (sum) of all such contributions in the material: the original wave plus the waves radiated by all the moving charges. This wave is typically a wave with the same frequency but shorter wavelength than the original, leading to a slowing of the wave's phase velocity. Most of the radiation from oscillating material charges will modify the incoming wave, changing its velocity. However, some net energy will be radiated in other directions or even at other frequencies (see scattering).

Depending on the relative phase of the original driving wave and the waves radiated by the charge motion, there are several possibilities:

If the electrons emit a light wave which is 90° out of phase with the light wave shaking them, it will cause the total light wave to travel slower. This is the normal refraction of transparent materials like glass or water, and corresponds to a refractive index which is real and greater than 1.^[26]^{[page needed]}
If the electrons emit a light wave which is 270° out of phase with the light wave shaking them, it will cause the wave to travel faster. This is called "anomalous refraction", and is observed close to absorption lines (typically in infrared spectra), with X-rays in ordinary materials, and with radio waves in Earth's ionosphere. It corresponds to a permittivity less than 1, which causes the refractive index to be also less than unity and the phase velocity of light greater than the speed of light in vacuum $c$ (note that the signal velocity is still less than $c$ , as discussed above). If the response is sufficiently strong and out-of-phase, the result is a negative value of permittivity and imaginary index of refraction, as observed in metals or plasma.^[26]^{[page needed]}
If the electrons emit a light wave which is 180° out of phase with the light wave shaking them, it will destructively interfere with the original light to reduce the total light intensity. This is light absorption in opaque materials and corresponds to an imaginary refractive index.
If the electrons emit a light wave which is in phase with the light wave shaking them, it will amplify the light wave. This is rare, but occurs in lasers due to stimulated emission. It corresponds to an imaginary index of refraction, with the opposite sign to that of absorption.

For most materials at visible-light frequencies, the phase is somewhere between 90° and 180°, corresponding to a combination of both refraction and absorption.

Dispersion

A rainbow — Light of different colors has slightly different refractive indices in water and therefore shows up at different positions in the rainbow.

A white beam of light dispersed into different colors when passing through a triangular prism — In a triangular prism, dispersion causes different colors to refract at different angles, splitting white light into a rainbow of colors. The blue color is more deviated (refracted) than the red color because the refractive index of blue is higher than that of red.

The refractive index of materials varies with the wavelength (and frequency) of light.^[27] This is called dispersion and causes prisms and rainbows to divide white light into its constituent spectral colors.^[28] As the refractive index varies with wavelength, so will the refraction angle as light goes from one material to another. Dispersion also causes the focal length of lenses to be wavelength dependent. This is a type of chromatic aberration, which often needs to be corrected for in imaging systems. In regions of the spectrum where the material does not absorb light, the refractive index tends to decrease with increasing wavelength, and thus increase with frequency. This is called "normal dispersion", in contrast to "anomalous dispersion", where the refractive index increases with wavelength.^[27] For visible light normal dispersion means that the refractive index is higher for blue light than for red.

For optics in the visual range, the amount of dispersion of a lens material is often quantified by the Abbe number:^[28]

V={\frac {n_{\mathrm {yellow} }-1}{n_{\mathrm {blue} }-n_{\mathrm {red} }}}.

For a more accurate description of the wavelength dependence of the refractive index, the Sellmeier equation can be used.^[29] It is an empirical formula that works well in describing dispersion. Sellmeier coefficients are often quoted instead of the refractive index in tables.

Principal refractive index wavelength ambiguity

Because of dispersion, it is usually important to specify the vacuum wavelength of light for which a refractive index is measured. Typically, measurements are done at various well-defined spectral emission lines.

Manufacturers of optical glass in general define principal index of refraction at yellow spectral line of helium (587.56 nm) and alternatively at a green spectral line of mercury (546.07 nm), called $d$ and $e$ lines respectively. Abbe number is defined for both and denoted $V d$ and $V e$ . The spectral data provided by glass manufacturers is also often more precise for these two wavelengths.^[30]^[31]^[32]^[33]

Both, $d$ and $e$ spectral lines are singlets and thus are suitable to perform a very precise measurements, such as spectral goniometric method.^[34]^[35]

In practical applications, measurements of refractive index are performed on various refractometers, such as Abbe refractometer. Measurement accuracy of such typical commercial devices is in the order of 0.0002.^[36]^[37] Refractometers usually measure refractive index $n D$ , defined for sodium doublet $D$ (589.29 nm), which is actually a midpoint between two adjacent yellow spectral lines of sodium. Yellow spectral lines of helium ( $d$ ) and sodium ( $D$ ) are 1.73 nm apart, which can be considered negligible for typical refractometers, but can cause confusion and lead to errors if accuracy is critical.

All three typical principle refractive indices definitions can be found depending on application and region,^[38] so a proper subscript should be used to avoid ambiguity.

Complex refractive index

When light passes through a medium, some part of it will always be absorbed. This can be conveniently taken into account by defining a complex refractive index,

{\underline {n}}=n+i\kappa .

Here, the real part $n$ is the refractive index and indicates the phase velocity, while the imaginary part $κ$ is called the optical extinction coefficient or absorption coefficient—although $κ$ can also refer to the mass attenuation coefficient^[39]^: 3—and indicates the amount of attenuation when the electromagnetic wave propagates through the material.^[1]^: 128

That $κ$ corresponds to absorption can be seen by inserting this refractive index into the expression for electric field of a plane electromagnetic wave traveling in the $x$ -direction. This can be done by relating the complex wave number $k$ to the complex refractive index $n$ through $k = 2π n / λ 0$ , with $λ 0$ being the vacuum wavelength; this can be inserted into the plane wave expression for a wave travelling in the $x$ -direction as:

{\begin{aligned}\mathbf {E} (x,t)&=\operatorname {Re} \!\left\\&=\operatorname {Re} \!\left\\&=e^{-2\pi \kappa x/\lambda _{0}}\operatorname {Re} \!\left.\end{aligned}}

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